Question

1. Consider the general form of the utility for goods that are perfect complements.
a) Why won’t our equations for finding an interior solution to the consumer’s problem work for this kind of utility? Draw(but do not submit) a picture and explain why (4, 16) is the utility maximizing point if the utility is U(x, y) = min(2x, y/2), the income is $52, the price of x is $5 and the price of y is $2. From this picture and explanation, can you describe where on the indifference curve the utility maximizing point is for any budget for this utility? (1 point)

b) Use the last part of (a) help you figure out how to write an equation which will contain all the points you described. Using this equation and the budget in (a), solve for the utility maximizing bundle and show that (4, 16) turns out to be the solution. (1 point)

c) Repeat part (b) but now use a generic budget pxx + pyy = I instead of the specific prices and income to find the demands for good x*(px,py,I) and good y*(px,py,I) associated with the utility is U(x, y) = min(2x, y/2). Plug in the prices and income from

**
(b) to verify that the functions gives you the correct (x , y ) in the one case where you

know the answer. (1 point)
2. A consumer is choosing between 3 goods, (x1, x2, x3). The consumer’s utility is

1⁄4 1⁄2 1⁄4
?(?1,?2,?3) = ?1 ?2 ?3 . The consumer has I = 128 and faces prices P1 = 2, P2 = 16

and P3 = 32.
a) Write out the Lagrangian for the consumers problem and find the equations for the first order conditions to a solution. (1 point)

b) Use the first order conditions to verify that x1=16, x2=4 and x3=1 is the solution to the consumer’s problem. What is the additional utility from an extra $ for this consumer in this situation? (1 point)

12 3.GivenaU(x,y)=x3y3,Px =1,Py =2,andI=96:

a) Find the optimal (x,y) and the utility from that bundle. (1 point)
b) Find what happens when you change Px = 8 but keep Py = 2 two different ways:

i) Keep the income = 96
ii) Keep the utility at the same level you found in (a)

Do these two methods give you the same answer? Does this make sense? Explain. (2 points)

c) How much does the answer to (b) part (ii) cost at the prices in part (b)? Is it more or less than 96? Does that make sense? (1 point)

d) For a general px, py, and I or U depending on the problem, find x * ( p x , p y , I ) and xc (px , py ,U). (Note: For this problem, don’t need either of the demands

for y) (1 point)

2
4. Answer the following based on a utility function U(x, y) = x3 + y , P = 2 and P = 9 .

a) If income = 90, find the optimal quantity of the two goods. Now change income to 48 and try to solve this problem using the general method. Explain why the solution you get cannot be correct and find the actual solution to the problem. (note: a picture may be helpful but don’t submit it.) (1 point)

b) Find the demand functions for x*(px,py,I) and y*(px,py,I) assuming both x>0 and y>0. (1 point)

c) Explain from these functions why it will always be that x>0, but y>0 only if a condition is met. Show that condition. Also show the demand functions for x and y if the condition is not met. Explain how this result is consistent with part a. (1 point)

d) Find the indirect utility function for the case where y>0 and for where y=0. (1 point)

e) Find the minimum expenditure associated with px=2, py=9 and a utility of 13. Explain why this answer is consistent with earlier results. (1 point)

Answer 1

I have answered the first question for you. I hope it helps.

Answer 1:

(a)The Indifference curve for a perfect complement case comes in the form of an L-shaped curve. Due to this kink, the function is non-differentiable and hence, the use of lagrange is not fruitful since lagrange method requires you to differentiate the optimization function. That is why we can't find an interior solution using lagrange.

I have plotted the ICs for the given utility function and the budget line, 5x + 2y = 52, on a graph in the picture below to show that the optimum bundle is (4,16).

We plot the budget line using the Intercept-form of line equation as shown in the text picture above. The ICs for the given utility functions are the L -shaped level curves that will have kinks at the point 2x=y/2 => 4x=y.

The utility maximizing bundle is the one where the individual maximizes his/her utility function given the budget constraint, i.e. the point where the indifference curve is tangent to the budget line. In the picture above, that point is (4,16) where the IC1 (One IC out of the many from the family of ICs of the utility function U = min {2x,y/2}) is tangent to the budget line 5x + 2y = 52.

(b) Now, we have two equations, 4x=y ............(1)

5x + 2y = 52 ......................(2)

Substituting (1) in (2) gives us: 5x + 2*4x = 52 => 13x=52 => x*=4

And, y=4x => y=4*4 => y*=16

This gives us the utility maximizing bundle, (x*,y*) = (4,16).

(c) Now, we have two equations, 4x=y ..................(3)

Px*x + Py*y = I .................(4)

Substituting (3) in (4) gives us : Px*x + Py*4x = I => x (Px+4Py) = I

=> x (Px,Py,I) = I / (Px+4Py) {Demand function for x}

And,   y (Px,Py,I) = 4* x = 4*I / (Px+4Py) {Demand function for y}

Now, for I=$52, Px=$5, and Py=$2, we get: x = 52/ (5+4*2) = 52/13 = 4

y = 4*52/(5+4*2) = 208/13 = 16

Thus, we get our utility optimizing bundle (4,16) back.